Sequences

To find the seventh term, substitute n=7 in the rule: (a_7=4(7)^2-3(7)=4×49-21=196-21=175). Hence, the correct answer is 175. The value 196 is only 4×49; subtracting 3×7 is also necessary. Exam tip: evaluate the power first, then multiply and subtract.

Given \(a_n=n^3-2n\), substitute \(n=5\): \(a_5=5^3-2(5)=125-10=115\). Therefore, 115 is correct. A value such as 120 can result from not subtracting \(2n\) correctly. Exam tip: evaluate powers first, then multiplication, and finally subtraction.

Each term in the sequence is 9 greater than the previous term: 9, 18, 27, 36, 45. Therefore, the term after 27 is \(27+9=36\). Choosing 34 or 38 would not maintain the common difference of 9. Exam tip: Find the difference between consecutive terms to identify the pattern.

Check the differences between the given terms: \(152-138=14\) and \(138-124=14\). Thus, each successive term is 14 less than the previous term. Therefore, \(180-14=166\), so the blank is 166. If 164 were chosen, the next difference would be 12, so it would not maintain the pattern. Exam tip: In a missing-term sequence, first check the differences between consecutive given terms.

The terms do not increase by a fixed amount, so we examine their consecutive differences. They are \(12-5=7\), \(22-12=10\), \(35-22=13\), and \(51-35=16\). These differences are 7, 10, 13, 16, and each is 3 greater than the previous one. This gives a clear rule for the next step.

The next difference is \(16+3=19\). Adding this to the fifth term gives \(51+19=70\). Therefore option B is correct. The sequence can also be understood as having second differences equal to 3, but calculating the first differences is enough here. The answer is not obtained by adding a constant number directly to the original terms.

The amounts subtracted are \(5, 10, 15, 20\). Each subtraction increases by \(5\), so the next subtraction is \(25\). Therefore, the next term is \(200-25=175\). \(170\) would result from subtracting \(30\), which does not follow the pattern. Exam tip: For next-term questions, first check the differences between consecutive terms.

This is a geometric sequence in which each term is 4 times the preceding term. After the given terms, the 5th term is 384 × 4 = 1536, and the 6th term is 1536 × 4 = 6144. Therefore, 6144 is correct. The closest distractor, 1536, is the 5th term, not the 6th. Exam tip: identify the common multiplier to find subsequent terms quickly.

This is a geometric sequence in which each term is one-third of the preceding term: \(729\div3=243\), \(243\div3=81\), and \(81\div3=27\). Therefore, the next term is \(27\div3=9\). Neither 6 nor 12 follows the same division rule. Exam tip: Compare consecutive terms to identify the pattern or common ratio.

The given terms are consecutive perfect squares: \(16=4^2, 25=5^2, 36=6^2, 49=7^2\), and \(64=8^2\). Hence, the \(n\)th term is \((n+3)^2\). Therefore, the \(8\)th term is \((8+3)^2=11^2=121\). Note that \(144=12^2\) is the next, or \(9\)th, term. Exam tip: For such sequences, take square roots of the terms to identify the underlying number pattern.

The terms are \(4^3, 5^3, 6^3, 7^3, \ldots\), respectively. Hence, the fifth term is \(8^3=512\), while the sixth term is \(9^3=729\). Therefore, 729 is correct. In exams, first identify the sequence of base numbers and then apply the exponent.

In option A, consecutive differences are \(7-3=11-7=15-11=4\), so it is an AP. Its ratios are not constant, hence it is not a GP. Exam tip: check differences before ratios.

By the given rule, each new term is the sum of the two terms immediately before it: 14 = 4 + 10, 24 = 10 + 14, and 38 = 14 + 24. Therefore, the next term is 24 + 38 = 62. An option such as 60 does not equal the sum of the previous two terms. Exam tip: for this type of sequence, add the last two given terms to find the next term.

This is a geometric sequence in which each term is obtained by multiplying the previous term by 3: 9, 27, 81, 243, 729. Therefore, 729 is the 5th term. The 4th term is 243, so it is not correct. Exam tip: List the terms in order and check for a common multiplier.

This is an arithmetic sequence with first term 15 and common difference 9. Therefore, the nth term is \(a_n=15+(n-1)\times 9\). Putting \(a_n=105\), \(15+(n-1)\times9=105\), so \(n-1=10\) and \(n=11\). Hence, 105 is the 11th term. The 10th term is 96, so it is not correct. Exam tip: use \(a_n=a+(n-1)d\) to find the position of a term in an arithmetic sequence.

This is an arithmetic progression with first term 170 and common difference \(-12\). Therefore, \(a_n=170+(n-1)(-12)\). Setting \(a_n=74\) gives \(170-12(n-1)=74\), so \(n-1=8\) and \(n=9\). Hence, 74 is the 9th term. The 8th term is 86, so it is not correct. Exam tip: To find the position of a term, substitute its value in \(a_n=a+(n-1)d\).

The differences between consecutive terms are \(20-9=31-20=42-31=11\), so this is an arithmetic progression. Its nth term is \(a_n=a_1+(n-1)d=9+(n-1)\times11=11n-2\). In option B, putting \(n=1\) gives 20 as the first term, so it is incorrect. Exam tip: To check a linear rule quickly, substitute \(n=1\) and then \(n=2\).

This is an arithmetic sequence because each successive term increases by 9. Here, \(a_1=16\) and \(d=9\). Therefore, \(a_n=a_1+(n-1)d=16+9(n-1)=9n+7\), so option A is correct. In option B, substituting \(n=1\) gives 25 as the first term, not 16. Exam tip: Check a proposed \(a_n\) rule by putting \(n=1\) to verify the first term.

Substituting \(n=1,2,3,4\) in the rule gives \(a_1=8(1)-5=3\), \(a_2=11\), \(a_3=19\), and \(a_4=27\). Therefore, the first four terms are \((3,11,19,27)\). The option \((8,16,24,32)\) uses only \(8n\) and ignores the subtraction of 5. Exam tip: start with \(n=1\) to find the first term.

Substitute n=6: a_6=3(6^2)+4(6)=3(36)+24=108+24=132. Therefore, the correct answer is 132. An option such as 128 can result from an error in squaring or multiplication. Exam tip: evaluate powers first, then multiply, and finally add.

To find the eighth term, substitute n=8 in the rule: \(a_8=140-9(8)=140-72=68\). Hence, 68 is correct. 66 may result from an incorrect subtraction of \(140-72\), while 72 is only the value of \(9\times8\), not the term itself. Exam tip: for an nth-term rule, multiply first and then subtract.

Given \(a_n=7n^2\), substitute \(n=1,2,3,4\). This gives \(7\times1^2=7\), \(7\times2^2=28\), \(7\times3^2=63\), and \(7\times4^2=112\). Hence, the sequence is \(7, 28, 63, 112\). Option A follows \(7n\), not \(7n^2\). Exam tip: For a rule-based sequence, substitute the first few natural-number values of \(n\).

For the first four terms, take n=1,2,3,4. Thus, a_1=3^1+1=4, a_2=3^2+2=11, a_3=3^3+3=30, and a_4=3^4+4=85. Therefore, the correct sequence is (4,11,30,85). Option B contains only powers of 3 and does not add n. Exam tip: evaluate 3^n first and then add n for each term.

Both sequences have common difference (7), so the term difference always remains (4). The (12)th terms also differ by (4).

Both sequences have difference (8), and their first terms differ by (9). So their (9)th terms also differ by (9).

This is an arithmetic progression with first term \(a=8\), common difference \(d=8\), and \(n=9\) terms. Using \(S_n=\frac{n}{2}[2a+(n-1)d]\), we get \(S_9=\frac{9}{2}[16+8\times8]=\frac{9}{2}\times80=360\). Hence, 360 is correct. The value 384 may result from using an incorrect number of terms or last term. Exam tip: identify \(a\), \(d\), and \(n\) before applying the sum formula.